Moreover, we know that \(\mu _{g}+\rho ^L+ \rho ^R\)  is a Maurer-Cartan element in the graded Lie algebra \(\oplus _n\mathrm {Hom}(({g}\oplus {h})^{\otimes n}, {g}\oplus {h})\)  from [1]}, Thus it induces a differential \(d_{\mu _{g}+\rho ^L+ \rho ^R}=\llbracket \mu _{g}+\rho ^L+ \rho ^R, ~\rrbracket \) , and the graded space \(\oplus _n\mathrm {Hom}({g}^{\otimes n}, {h})\)  is closed under the differential \(d=d_{\mu _{g}+\rho ^L+ \rho ^R}\)  and is given by
\(&(d f)(x_1,\ldots , x_{n+1})\\&=(-1)^{n+1}\sum ^{n}_{i=1}(-1)^{i+1}\rho ^L(x_i)f(x_1,\ldots , \hat{x}_i, \ldots , x_{n+1}) + \rho ^R(x_{n+1})f(x_1,\ldots , x_{n})\\&+(-1)^{n+1}\sum _{1\le i< j\le n+1} (-1)^{i}f(x_1,\ldots , \hat{x}_i, \ldots , x_{j-1}, [x_i, x_j]_{g}, x_{j+1},\ldots , x_{n+1}).\) 
